3.1. Modelling the Indirect Utility Function with Rational Agents
Filippini and Hunt [
3] argue that inefficiency arises if there is excessive use of energy. Our purpose is to model the efficiency and inefficiency of the use of residential electricity, but electricity is not itself a final commodity or a good that enters the utility function of the consumer. It is an input that permits the constrained optimal consumption of the allocation of final goods and services. This idea goes back at least to the contributions of Lancaster [
16]. Grösche [
17] argued that residential energy consumption derives from the demand for energy services, such as the demand for thermal comfort. Households produce those services with their energy commodities (e.g., heating equipment) by using a set of fuel inputs.
We assume that the final goods and services consumed by the household are represented by the vector with corresponding nominal prices: . The consumer’s nominal income is symbolized as , to differentiate it from real income. However, in order to be able to consume one or more of the final goods and services, technology dictates that the consumer must use a minimum amount of input of electrical energy in kWh, . Complementary to the required input of electrical energy, there is a required power capacity in kW or kVA, which we label .
The household’s minimum required electrical energy,
is determined by a number of exogenous factors that include household characteristics and assets. Often in residential energy demand studies, these factors are related to family size and the characteristics of the building which the household occupies. However, the building itself is likely to be an indirect measure of what determines energy demand. In the case of electrical energy demand, a much more direct measure is the nature of the appliances and equipment used by the household. In fact, two different households occupying identical buildings may have relatively different electrical energy demands if they hold different ranges of appliances or there are different numbers of occupants. We use
to represent a vector of these exogenous factors including family size and the ownership of different types of appliances and equipment with different power loadings. Hence, we write frontier electrical energy input,
, as
There will be a complementary household subscribed power demand in kW or kVA since power demand
is a non-decreasing function of energy usage:
. The complementary input of power capacity, consistent with the provision of the minimum input of electrical energy to sustain a particular living standard, is also determined by exogenous factors including household characteristics and assets which we label
. Therefore, frontier power demand is
.
The minimum amount of the input of electrical energy, , is bought at the market price, which typically in European markets has two components: an energy charge per unit of electrical energy, (kWh) consumed, , and a monthly subscribed demand charge, , which is calibrated to the level of subscribed demand (kW or kVA) corresponding to each individual consumer. The household is able to choose from a menu of demand charges by selecting a specific subscribed demand which is restricted by circuit breakers in the property. This is the case in the data that we study here. However, in some cases, e.g., in the UK, and parts of the USA, this is set for all residential consumers at a fixed level determined by the retail supplier or utility offering the price, and is referred to simply as the standing charge.
The consumer’s budget constraint is therefore:
In Equation (5),
represents the expenditure on goods,
is nominal income,
is the nominal energy price,
is energy consumption,
is the nominal demand charge,
and is the subscribed power demand. The solution to the consumer’s utility maximization problem given the existence of the vectors of exogenous conditioning variables,
and
produces the indirect utility function:
The solution to this consumer problem is analyzed in
Appendix A, where the first-order conditions are shown to be:
Each of the prices is set equal to the marginal benefit to the consumer of the product or the electrical energy input or power input expressed in terms of a money metric of individual welfare given by the marginal indirect utility of money income.
The demand functions derived from the interior optimum with binding constraints are
Equation (10) shows the usual consumer demand functions for goods and services entering the utility function, while Equations (11) and (12) show the demands for electrical energy and power capacity or the subscribed demand conditional on the nominal price of all goods, energy charges, demand charges, money income and the exogenous variables that determine the technological requirement for electrical energy. The consumer’s net income available to spend on final goods and services is
, i.e., the money income minus the expenditure on the energy input comprising energy charges and demand charges. If the consumer’s actual use of energy, exceeds the efficient minimum required, if
, the net income available for final goods and services is reduced and therefore utility, which is increasing in
in the indirect utility in Function (11), is reduced as well:
Following Equations (1) and (2), define the ratio of minimum electrical energy requirement to the actual electrical energy used as behavioral efficiency,
, then,
The inverse of behavioral efficiency, , therefore measures the extent to which the household’s electricity consumption exceeds the level that would maximize the rational consumer’s welfare at the same prices, income levels, family characteristics and appliance ownership.
Now, write
as an exponential function of the random error component ‘electrical energy inefficiency’,
, where
, so that:
Now, using the demand function for energy at the interior optimum (11), we replace the expression
to obtain:
In this way, we are able to interpret household behavior towards energy efficiency against the benchmark of an indirect utility frontier, where the frontier function is the demand function for the minimum energy which is required to enable the consumer to enjoy the utility from the consumption of final goods.
There is a similar analysis for the case where the consumer’s subscribed demand exceeds the minimum efficiency required, so that we have the analogous expression for the behavioral efficiency of the subscribed power demand:
Applying the previous argument:
Taking logs and adding an idiosyncratic error component,
, we have the familiar stochastic frontier analysis of the energy demand function with the difference that we do not need to assume that
yields direct utility to the consumer.
Using
, the personal consumption deflator to represent the vector of the prices of final goods and services,
, we will represent the demand function in practice as:
In (20), energy demand is a function of the energy charge in constant prices, the net income of the household after the payment of demand charges again in constant prices and the vector of exogenous variables,
, that shift the household energy demand function. Then, the equation to be estimated for household electrical energy demand is
This is our version of the Filippini and Hunt demand function for residential energy. We note that our derivation emphasizes individual household behavior and treats the issue of multiple goods and services directly. The inefficiency component is a random error with a positive mean . Consequently, in the standard stochastic frontier analysis, it is fitted as a ‘cost’ function. The fitted function will be in either double-log form or translog form.
The energy consumption in kWh given by Equation (20) will in turn determine a household subscribed power demand in kW or kVA since power demand
is a non-decreasing function of energy usage:
. Therefore, we arrive at a complementary subscribed power demand function:
In Equations (21) and (22) we distinguish the component error terms and , and the vectors of exogenous variables which form the full set of exogenous variables in the model: .
Our primary interest is in the income elasticity of electrical energy demand, the price elasticity of electrical energy demand, and the extent of behavioral efficiency for which we use the mean behavioral efficiency from the frontier estimation,
. However, it is critical to note that the price to the consumer has two components, the energy charge (in Euros per kWh in our dataset),
, and the subscribed demand or standing charge (in Euros per month in our dataset). The elasticity definitions are shown in
Table 2. In the case of appliances, define the percentage effect as the 100
the proportionate or relative change in energy demand when the ownership or installation status of the appliance changes from zero to one.
3.2. Amending the Model for Behavioral Agents and Overconsumption
In our study, we must also address the questions of why the consumer’s actual use of energy and subscribed power demand may exceed the efficient minimum. We explore the contribution of behavioral economics to this issue. Decker [
18] provides a wide-ranging survey of results in behavioral economics while Farhi and Gabaix [
19] offer a detailed model of the contrast between the indirect utility functions of rational and behavioral agents. Decker distinguishes between standard and non-standard consumers, while Farhi and Gabaix refer to rational and behavioral agents. The standard consumer or rational agent displays the rational calculating behavior that we illustrated in deriving the demand functions in Equations (11) and (12). However, the non-standard consumer or behavioral agent may not have stable or consistent preferences, may display various biases in the expectation of future utility, may be unwilling to actively compare different product offerings and may not be a utility-maximizer. Farhi and Gabaix identify the issues of internality and hyperbolic discounting which may be particularly important in energy demand. Internality refers to the long-term benefit or cost to the agents themselves that they fail to consider when deciding to consume a good or service. Hyperbolic discounters, according to Laibson [
20], use a high discount rate over short horizons and a low discount rate over long horizons, so that there is a conflict between today’s preferences and the preferences that will be held in the future. This leads to the procrastination of efficient decisions. These biases are observed in laboratory simulations but also in interviews with focus groups of energy consumers, according to Hallin et al. [
21].
Applying the analysis in Farhi and Gabaix (2020), we amend our rational consumer model as follows. The behavioral agent is assumed to maximize a perceived utility function to make consumption and energy input decisions that satisfy the budget constraint in Equation (5) but that also incorporate behavioral biases so that they do not fulfill the other optimality conditions in Equations (7)–(9) that apply to the rational consumer. Instead, Farhi and Gabaix introduce the concept of a behavioral wedge. This is expressed by the difference between the price of a commodity and the marginal utility (expressed in a money metric, as captured by
). The behavioral wedge is zero for the rational agent but nonzero for the behavioral agent. Therefore, the demand functions for each commodity and the inputs of energy and power demand satisfy:
For example, if the behavioral agent consumes excess electrical energy, this will be shown by the money metric marginal indirect utility for energy being below the market price of energy. The divergence of the money value of marginal utility from the market price reflects the failure of the agent to internalize all the costs of their choice. Therefore, the key to the measurement of
and
is the extent to which the agent is estimated not to internalize all the costs of their consumption decisions. In Equations (17)–(20), we have shown how this this will appear as behavioral inefficiency. Therefore, we write:
Gruber and Koszegi [
22], also cited by Farhi and Gabaix, use a similar approach to model smokers who are hyperbolic discounters.
A key element of energy policy is the effort to restrain energy consumption and power demand by price signals, e.g., those that incorporate a carbon emissions tax in the price of energy and power demand to consumers. Farhi and Gabaix demonstrate that the welfare effects of such price changes can be measured by amending the measures of the welfare effects of price changes given by Roy’s identity for the presence of the behavioral wedge. As demonstrated in
Appendix A, for the rational agent, for a small price change, the money metric of the loss of welfare or real income or utility is approximately equal to the change in price multiplied by the negative of the quantity demanded:
For the behavioral agent, the welfare effects of price changes in the amended Roy’s identities are:
In Equations (31) and (32), the terms
and
are the Slutsky substitution effects of the price changes on the consumption of energy and power, for example,
when the income effect of the price change has been compensated. In general, we expect own price and complementary price Slutsky effects to be negative so that the net effect is to reduce the welfare loss to the behavioral agent of a rise in price. A full derivation of these results is described in the work of Farhi and Gabaix [
19], but an intuitive explanation can be given by thinking of a consumer’s two-dimensional convex indifference curve diagram for true or experienced utility, as illustrated in
Figure 1. The rational agent is located at A at the tangency of an indifference curve with the budget constraint line,
, thereby fulfilling all the first-order conditions for a constrained utility maximum. The behavioral agent is located at an intersection of the budget line and a convex experienced utility indifference curve rather than at a tangency, fulfilling only the budget constraint condition, so that they display the overconsumption of one of the commodities. For example, the behavioral agent may be located at B on the budget constraint
, where there is an overconsumption of commodity
. If the price of this commodity rises so that the budget constraint pivots inwards to
, the behavioral agent will likely substitute away from the commodity by reducing overconsumption, locating at a point C further along the new budget line, closer to the rational agent optimum. But this represents a higher level of experienced utility shown by the dashed indifference curve through C, since the previous allocation was at a more extreme intersection of the budget line and the convex indifference curve. Clearly, the extent to which the behavioral agent can move to a higher experienced utility indifference curve as a result of the price rise reflects the substitution effect in the demand function and, hence, this enters the amended Roy’s identity for the welfare effects of a price change for the behavioral agent. Thus, a price rise for energy or power demand will include a positive offset on the welfare of behavioral agents because they are incentivized to be located closer to the rational agent’s optimum, i.e., to bring their consumption closer to the efficient frontier. The results shown later in this paper will be related to these comparative static effects shown in
Figure 1.
To extract the welfare effects of a price change, we use the well-known Slutsky equations in an elasticity format:
This theoretical analysis suggests that Pigouvian taxes might be used to correct for overconsumption and behavioral inefficiency. However, Farhi and Gabaix recognize two objections to this policy. First, in the model, agents make mistakes that the government can identify, but correcting for these to move consumers closer to the rational agent optimum is a form of paternalism. Additionally, governments may not understand agents’ motives and constraints well and may find it politically difficult to implement taxes to improve behavioral decisions. Second, if poorer consumers are the ones making behavioral mistakes such as internality or hyperbolic discounting, then these taxes may worsen the income distribution, which they describe as a trade-off between internality correction and redistribution. As well as instigating regulated price and tax changes, policy makers have the option of direct controls and what have become known as nudge policies, i.e., those that guide consumer decisions through better information and advice, and provide guidance on making efficient decisions. Farhi and Gabaix [
19] argue that these are particularly relevant where external effects are present, for example, in the case of a policy to restrict carbon emissions. In the empirical section of this paper, we attempt to estimate these effects statistically.
3.3. Estimation of Behavioral Efficiency
The aim is to estimate Equations (21) and (22) but there are several options available. Our estimation strategy objective is to obtain consistent and efficient estimators of the parameters including the regression elasticities and the mean and variance parameters of the composed error distributions. Each of the proposed estimation strategies does this under different assumptions about the covariance of the error terms.
We begin by treating the two equations for energy demand and subscribed power demand as unrelated and using standard stochastic frontier analysis maximum likelihood estimation (MLE-SFA) applied to each equation separately. This is applied to two cross section samples for 2008 and 2014. It can also be applied to a pooled cohort pseudo-panel after testing for the poolability of the two cross section samples.
The general stochastic frontier analysis model for a cross section sample of
observations is the regression:
There are two components to the error term
:
is the usual regression two-sided symmetric zero-mean random variable reflecting sampling, specification and measurement error, while
is the error component reflecting inefficiency, and is a one-sided asymmetrically distributed random variable with a positive mean in the case of the models used here. The inefficiency component is based on the idea that inefficiency can be treated as a nonnegative random variable with most observations clustered close to the positive side of the zero value and a positive tail of observations reflecting greater values of inefficiency with diminishing probability. The original model is due to Aigner Lovell and Schmidt [
23] and Meeusen and van den Broeck [
24] and it has been extensively developed since then; see, for example, the studies conducted by Kumbhakar and Lovell [
25] and Kumbhakar et al. [
26]. The basic model makes the following distributional assumptions:
and
so that the log-likelihood function of the sampled
with respect to the parameters
can be constructed as follows, where
, and
and
is the cumulative distribution function of the standard normal distribution from the composed normal and half-normal error terms.
Maximizing this log likelihood function by numerical methods after substituting for the
from Equation (35) allows us to estimate the parameters and to test for the presence of inefficiency. If
does not differ significantly from zero, then this implies
and the inefficiency distribution collapses on the zero value, so we conclude inefficiency is absent. If
does differ significantly from zero, then this implies
, and inefficiency is present in the sample. We have already defined behavioral efficiency for energy consumption in Equation (15) as
. A measure of inefficiency can then be based on the conditional expected value from estimation:
; see the studies conducted by Jondrow et al. [
27] and Battese and Coelli [
28]. This is one of the core models that we estimated.
However, other econometric assumptions must also be allowed for. One alternative is to treat Equations (21) and (22) as a related pair of behavioral functions so that we adopt the seemingly unrelated regression model estimated by generalized least squares, GLS-SUR. This permits the composed error terms in the two equations to be interdependent, indicating that similar random shocks are applied to both energy consumption and to power demand. Then, the feasible iterative GLS-SUR procedure produces statistically consistent estimators for the parameters . We are able then to test cross-equation restrictions on the elasticities.
The standard MLE-SFA is no longer suitable, so that now we use the quasi-maximum likelihood estimation suggested by Fan et al. [
29]; see also the work of Henningsen and Kumbhakar [
30] and Kuosmanen et al. [
31] to decompose the residuals from the consistent GLS-SUR estimates. Since the GLS-SUR estimators have the consistency property, we use
as symbolic notation for the residuals from the GLS-SUR.
Fan et al. show that the quasi-likelihood can be written as a function of the single parameter
, the signal-to-noise ratio, after replacing the error terms
in Equation (36) with these residuals from the consistent estimator
as follows.
Use (38) and (39) to substitute out
and
from (37). Maximize (37) by a grid search over values of the single parameter
to find the quasi-likelihood estimate
that maximizes (37), and insert estimate
into (38) and (39) to obtain (new) estimates of
and
. Subsequently, calculate
and
. Therefore, by applying the feasible GLS-SUR estimator to the pair of energy demand and power demand functions, and subsequently operating the Fan et al. [
29] procedure, we can obtain a new set of behavioral efficiency scores.
A third approach recognizes that Equations (21) and (22) might form a simultaneous equation system in which the endogeneity of one or more explanatory variables arises, so that the explanatory variables are correlated with the error term. One variable in particular may be a problem, and that is the consumer’s income after demand charges since this might be determined by the consumer’s reaction to the published demand charges. To address this issue, we use instrumental variable estimation with two-stage least squares, IV-2SLS. Again, to measure behavioral efficiency, we use the quasi-maximum likelihood estimation suggested by Fan et al. [
29] to decompose the residuals from the consistent IV-2SLS model. Therefore, we apply Equations (38) and (39) again where, this time, the residuals from the consistent estimators
are those from applying the IV-2SLS estimator. An alternative procedure is the Karakaplan and Kutlu [
32] instrumental variables treatment of the MLE-SFA estimator, used by Romero-Jordan and del Rio [
9].
Several types of parameter estimates are produced for , , , , depending on the sample: the cross section for 2008, (XS08), the cross section for 2014, (XS14), or the pooled pseudo-panel (PP) from each of three estimation strategies: MLE-SFA, GLS-SUR, and IV-2SLS. We have computed the results from all three modeling approaches, and it turns out that the results are very similar across all three approaches so that in the text we report only one set of results in full but comment on all three sets of results when we report the elasticities and the behavioral efficiency scores.